Heuristics in Ancient Arabic and Chinese Mathematics and its use in textbooks Prof. Dr. Bernd Zimmermann from University of Jena at University of Xi‘an

Heuristics in Ancient Arabic and Chinese

Mathematics and its use in textbooks

Prof. Dr. Bernd Zimmermann

from University of Jena at University of Xi‘an

August 2002

B. Zimmermann ICM Beijing 2002

Heuristics:

Methods to find conjectures

Methods to find proofs

Methods to (re)invent mathematics

By analysis of history one might find

methods/heuristics, which proved to be

most fruitful (“invariants”)


Example 1: Analogy

Archimedes

UR

? Kepler


Example 2: Analysis “Now, analysis is the path from what one is seeking, as if it

were established, by way of its consequences, to something that is established by synthesis.That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. We call this kind of method 'analysis', as if to say anapalin lysis (reduction backward).

In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. This is what we call 'synthesis'.” (Pappos in Jones A. (ed. &. transl.): Pappus of Alexandria. Book 7 of the Collection. Part 1. Springer, New York 1986. 1986, p. 82)


Ibn al Haitham, the method of analysis and perfect numbers

Jaouiche, K.: Ibn al Haitham: Kitab at-tahlil wa-t-

tarkib. Ouvrage d’al-H,.asan ibn al al-H,.asan ibn al

Haitham sur l’analyse et la synthèse. Unpublished

manuscript Paris 1991.

Rashed, R.: Ibn al-Haytham et les nombres parfaits.

In: Historia Mathematica 16 (1989), 343-352.

Hogendijk, J. P.: Review of Rashed 1989,

Mathematical Reviews Sections, 91d:01002 01A30

01A20 11-03, S. 1822, April 1991-Issue 91d.



Euclid Prop. 36: “If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, than the number is perfect.” (Heath T. L.: The Thirteen Books of Euclid’s Elements. Cambridge University Press, Cambridge 1925. Vol. 2, p. 421)

Modern form: If m=(1+2+22+23+…+2n)2n

and (1+2+22+23+…+2n)[=(2n+1-1)] is prime, than m is perfect.



Starting point of analysis:

Given an(y) even perfect number. What structure might it have?A. H.’s goal was not the conversion of the theorem of Euclid, but its heuristic foundation!

A. H. tries to generalize the experience of the “analysis” of the example 496=1+2+22+23+24+31+62+124+248

=(25-1)+31(1+2+22+23)=(25-1)(1+24-1) = (25-1)24


Ibn Sinan and heuristics

Bellosta, H.: Ibrahim ibn Sinan: On Analysis and Synthesis. In: Arabic Sciences and Philosophy, vol. I (1991), pp. 211 - 232

Content: Classification of problems; analysis and its role in the determination of the class of each problem; synthesis; reaction to criticism


Ibn Sinan and heuristics Example of a

problem. “Viviani’s” theorem:

In any equilateral triangle the sum of the distances from a point P within the triangle from all three sides is always the same.


Al Sijzi and problem fields


Al Sijzi and problem fields “Move” A and B in

such a way out of or into the Thales-circle, that these points are symmetric to the center of this circle.

“Move” C on the old Thales-circle. What is A’C2+B’C2 ; A’C’2 + B’C’2 ?

A B

C

B‘

A‘

C‘


Al Sijzi and problem fields


Heuristics from ancient China- applied in a German textbook

volume of a sphere


Some questions about occurrence of heuristics in ancient China

What about other testimonies concerning use of heuristic methods in ancient China?

In which way the results from the „Nine Chapters of Mathematical Technique“ or other famous ancient books were created?

Documents

Heuristics in Ancient Arabic and Chinese Mathematics and its use in textbooks Prof. Dr. Bernd Zimmermann from University of Jena at University of Xi‘an